This paper is concerned with the containment control problem for multi-unmanned aerial vehicle (multi-UAV) systems with data transmission among agents during a limited-bandwidth network. A good deal of data transmitted during a limited-bandwidth network may result in data collision, which brings negative effects on controller design and containment tracking implementation. To reduce the network load during data transmission, a consensus condition and a triggering subsidiary condition are first proposed to realize the containment tracking with H performance by using the triggered states. Then a self-triggered scheme is established under a multi-agent framework, in which the next triggering instant is calculated beforehand by using the past states. Combined with the self-triggered scheme, a containment controller is obtained via solving the derived consensus condition, which ensures that follower UAVs are driven into a convex set surrounded by leader UAVs. Finally, the validity of the self-triggered containment control method is proved through a simulation experience.